Quite, I struggled to formulate the example because indeed reading it the interpretation is that "them" could mean letters, and so doesn't sound completely as wrong as it does in the context of a person.

A better example would have been "I wrote Alice last week". Correct US English, utterly grating to British English. ( Technically still might not grate if your brain jumps to Alice being a Poem or other work of art! )

Because in British English we write letters, we don't write people. I don't know the term for it, it's not transitive vs intransitive, it's the verb object having a different restriction.